Strange Numbers

[Turtle]

I have found some previously undescribed numbers related to perfect numbers. Anyone care to comment? A few lists attached.

Addendum: The list of Strange Set elements is in post #146

[Tormod]

Originally Posted by Turtle

Sorry; I gave the wrong link:
http://home.comcast.net/~turtlediable/wsb/index.html

Hey Turtle - do you mind explaining to my non-mathematical mind what this is all about? I like number theory but didn't instantly get what these numbers are meant to represent.

[maddog]

Huh ?

Why is 24 strange. Why is it's sum(?) (to what ? 12?) strange. I'm missing it.

Maddog

[Bo]

i also don't see more then a way to generate a set of numbers... What's special?


Bo

[Turtle]

___Begin with the idea of perfect numbers;OK? Now if a number's divisor sum equals that number it's perfect, if the sum is less it's called deficient, & if the sum is more it's called abundant. So all Strange numbers are abundant by exactly 12(twice the first perfect number six) The set of Bizarre numbers abundant by exactly 56 (twice the second perfect number 28).
___The sets' discovery I believe is original to me; I have never seen them described. They are further special because most of the members of each set have the same number of divisors. Because some don't fit the general pattern of set elements having a perfect number as a divsor (304 for example in the Strange set), the only way to find these sets is to factor the integers sequentially. Since factoring is considered a "hard problem" in computer science, looking for large elements to the sets as well as elements that don't fit the general rule takes considerable time.
It's interesting because no one ever found them before & I did & the topic of perfect/deficient/abundant has been studied for thousands of years.
Lay it on me.

[maddog]

Originally Posted by Turtle

Begin with the idea of perfect numbers;OK? Now if a number's divisor sum equals that number it's perfect, if the sum is less it's called deficient, & if the sum is more it's called abundant. So all Strange numbers are abundant by exactly 12(twice the first perfect number six) The set of Bizarre numbers abundant by exactly 56 (twice the second number 28). The sets' discovery I believe is original to me; I have never seen them described. They are further special because most of the members of each set have the same number of divisors. Because some don't fit the general pattern of set elements having a perfect number as a divsor (304 for example in the Strange set), the only way to find these sets is to factor the integers sequentially. Since factoring is considered a "hard problem" in computer science, looking for large elements to the sets as well as elements that don't fit the general rule takes weeks.
It's interesting because no one ever found them before & I did & this topic has been studied for thousands of years.
Lay it on me.

I see I had forgot the definition of Perfect numbers, Thank You. So why does the Definition
of Strange numbers need to be abundant by 12 ??? So to hear you out..

The divisors of 12 are 1, 2, 3, 4, 6 in which the sum is 16 and thus abundant.
The divisors of 24 are 1, 2, 3, 4, 6, 12, in which the sum of 28 and thus abundant.

So this makes 24 and 36 strange because they are both 12 more than an abundant
number ? Am I getting this right ?

Maddog

[Turtle]

Almost there. What is critical here is the 'difference' between a number & the sum of its divisors. All the sets I found consist of abundant numbers. The uniqueness in them is the amount of their abundance;the difference. The quality of having a divisor sum exactly 12 more than itself is literally what defines "Strange Number".
To find all the Strange Numbers (actually the set is infinite), you have to factor the integers sequentially & subtract each integer from its divisor sum; when this difference is 12, voila, you found another Strange Number. If that difference is 56, you found a Bizarre Number: 992, found a Peculiar Number:16,256, found a Curious Number:etc.
Note that there are no other regular sets like this which have a difference of say 7, or 13, or 2,345, etc.

[maddog]

I would conjecture from your statement that every multiple of 12 beyond 12 is a strange
number using you definition.

To check the next few

36: 1, 2, 3, 4, 6, 9, 12, 18 => sum = 55 > 36 => abundant
48: 1, 2, 3, 4, 6, 8, 12, 16, 24 => sum = 76 > 48 => abundant
60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 => 108 > 60 => abundant

So what does this mean ???

Maddog

[Turtle]

___Ouch; just lost a twenty minute respose into the ether! Rats.
___Ok 60 is abundant but not Strange because 108-60=48. To be Strange the difference must be 12. You can find most Strange numbers by multiplying 6 (a perfect number) times a prime, but this does not find 304, which does not divide by 6(or 12). Most Strange numbers have exactly 8 divisors (4 pairs) & 304 has 10 (5 pairs) 304 I call an anomalous Strange Number & the next anomalous Strange Number is 127,744. The other sets also have these anomalous & so the only way to find them all is factoring.
___What does it mean? It means I found something everyone for a couple thousand years has overlooked. It's like if you found a new cave, you would explore on your own as much as possible & then begin showing people what you found. Let them decide what it means; but you always remain the ONE who found it first.

[maddog]

Originally Posted by Turtle

Ouch; just lost a twenty minute respose into the ether! Rats.
Ok 60 is abundant but not Strange because 108-60=48. To be Strange the difference must be 12. You can find most Strange numbers by multiplying 6 (a perfect number) times a prime. But this does not find 304, which does not divide by 6(or 12). Also most Strange numbers have exactly 8 divisors (4 pairs) & 304 has 10 (5 pairs) This why it's strange. 304 I call an anomalous Strange Number & the next one is 127,744. The other sets also have these anomolys & so the only way to find them all is factoring.
What does it mean? It means I found something everyone for a couple thousand years has overlooked. It's like if you found a new cave, you would explore on your own as much as possible & then begin showing people what you found. Let them decide what it means; but you always remain the ONE who found it first.

OK... So, why does difference "need" to be 12 ? (2 * 6 -- a perfect#)... ??? Couldn't
it be any other value ? What is so special about 12 ????

Maddog